Currency Options (2): Hedging And Valuation

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Currency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeOverviewChapter 9Currency Options (2):Hedging and Valuation

OverviewCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeOverviewThe Binomial Logic: One-period pricingThe Replication ApproachThe Hedging ApproachThe Risk-adjusted ProbabilitiesMultiperiod Pricing: AssumptionsNotationAssumptionsDiscussionStepwise Multiperiod Binomial Option PricingBackward Pricing, Dynamic HedgingWhat can go wrong?American-style OptionsTowards Black-Merton-ScholesSTP-ing of European OptionsTowards the Black-Merton-Scholes EquationThe Delta of an Option

OverviewCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeOverviewThe Binomial Logic: One-period pricingThe Replication ApproachThe Hedging ApproachThe Risk-adjusted ProbabilitiesMultiperiod Pricing: AssumptionsNotationAssumptionsDiscussionStepwise Multiperiod Binomial Option PricingBackward Pricing, Dynamic HedgingWhat can go wrong?American-style OptionsTowards Black-Merton-ScholesSTP-ing of European OptionsTowards the Black-Merton-Scholes EquationThe Delta of an Option

OverviewCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeOverviewThe Binomial Logic: One-period pricingThe Replication ApproachThe Hedging ApproachThe Risk-adjusted ProbabilitiesMultiperiod Pricing: AssumptionsNotationAssumptionsDiscussionStepwise Multiperiod Binomial Option PricingBackward Pricing, Dynamic HedgingWhat can go wrong?American-style OptionsTowards Black-Merton-ScholesSTP-ing of European OptionsTowards the Black-Merton-Scholes EquationThe Delta of an Option

OverviewCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeOverviewThe Binomial Logic: One-period pricingThe Replication ApproachThe Hedging ApproachThe Risk-adjusted ProbabilitiesMultiperiod Pricing: AssumptionsNotationAssumptionsDiscussionStepwise Multiperiod Binomial Option PricingBackward Pricing, Dynamic HedgingWhat can go wrong?American-style OptionsTowards Black-Merton-ScholesSTP-ing of European OptionsTowards the Black-Merton-Scholes EquationThe Delta of an Option

Binomial Models—What & Why?Currency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPractice Binomial ModelB given St , there only two possible values for St 1 , called “up” and“down”. Restrictive?—Yes, but .The Binomial Logic:One-period pricingMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial PricingTowardsBlackMertonScholesB the distribution of the total return, after many of these binomialprice changes, becomes bell-shapedB the binomial option price converges to the BMS priceB the binomial math is much more accessible than the BMS mathB BinMod can be used to value more complex derivatives thathave no closed-form Black-Scholes type solution. Ways to explain the model—all very similar:in spot marketforwardvia hedgingvia replication(not here)(not here)yesyes

Binomial Models—What & Why?Currency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPractice Binomial ModelB given St , there only two possible values for St 1 , called “up” and“down”. Restrictive?—Yes, but .The Binomial Logic:One-period pricingMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial PricingTowardsBlackMertonScholesB the distribution of the total return, after many of these binomialprice changes, becomes bell-shapedB the binomial option price converges to the BMS priceB the binomial math is much more accessible than the BMS mathB BinMod can be used to value more complex derivatives thathave no closed-form Black-Scholes type solution. Ways to explain the model—all very similar:in spot marketforwardvia hedgingvia replication(not here)(not here)yesyes

Binomial Models—What & Why?Currency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPractice Binomial ModelB given St , there only two possible values for St 1 , called “up” and“down”. Restrictive?—Yes, but .The Binomial Logic:One-period pricingMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial PricingTowardsBlackMertonScholesB the distribution of the total return, after many of these binomialprice changes, becomes bell-shapedB the binomial option price converges to the BMS priceB the binomial math is much more accessible than the BMS mathB BinMod can be used to value more complex derivatives thathave no closed-form Black-Scholes type solution. Ways to explain the model—all very similar:in spot marketforwardvia hedgingvia replication(not here)(not here)yesyes

OutlineCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingThe Replication ApproachThe Hedging ApproachThe Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial PricingTowardsBlackMertonScholesThe Binomial Logic: One-period pricingThe Replication ApproachThe Hedging ApproachThe Risk-adjusted ProbabilitiesMultiperiod Pricing: AssumptionsNotationAssumptionsDiscussionStepwise Multiperiod Binomial Option PricingBackward Pricing, Dynamic HedgingWhat can go wrong?American-style OptionsTowards Black-Merton-ScholesSTP-ing of European OptionsTowards the Black-Merton-Scholes EquationThe Delta of an Option

Our ExampleCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingThe Replication Approach DataB S0 INR / MTL100, r 5%p.p.; r 3.9604%. Hence:F0,1 S01 r0,11.05 100 101. 1 r0,11.039604B S1 is either S1,u 110 (“up”) or S1,d 95 (“down”).B 1-period European-style call with X INR/MTL 105C1The Hedging ApproachThe Risk-adjusted ProbabilitiesS1110Multiperiod Pricing:AssumptionsStepwise MultiperiodBinomial B slope of exposure line (exposure):exposure C1,u C1,d5 0 1/3S1,i S1,d110 95S3,3 S0uuuS S uuexposureline

Our ExampleCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingThe Replication Approach DataB S0 INR / MTL100, r 5%p.p.; r 3.9604%. Hence:F0,1 S01 r0,11.05 100 101. 1 r0,11.039604B S1 is either S1,u 110 (“up”) or S1,d 95 (“down”).B 1-period European-style call with X INR/MTL 105C1The Hedging ApproachThe Risk-adjusted ProbabilitiesS1110Multiperiod Pricing:AssumptionsStepwise MultiperiodBinomial B slope of exposure line (exposure):exposure C1,u C1,d5 0 1/3S1,i S1,d110 95S3,3 S0uuuS S uuexposureline

The Replication ApproachCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingThe Replication Approach Step 1 Replicate the payoff from the call—[5 if u] and[0 if d]:(a) forward contract(buy MTL 1/3 at 101)(b) deposit,V1 20(a) (b)S1 951/3 ( 95 101) 2 20S1 1101/3 (110 101) 3 25The Hedging ApproachThe Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial PricingTowardsBlackMertonScholes Step 2 Time-0 cost of the replicating portfolio:B forward contract is freeB deposit will costINR2/1.05 INR1.905 Step 3 Law of One Price: option price valueportfolioC0 INR1.905

The Replication ApproachCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingThe Replication Approach Step 1 Replicate the payoff from the call—[5 if u] and[0 if d]:(a) forward contract(buy MTL 1/3 at 101)(b) deposit,V1 20(a) (b)S1 951/3 ( 95 101) 2 20S1 1101/3 (110 101) 3 25The Hedging ApproachThe Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial PricingTowardsBlackMertonScholes Step 2 Time-0 cost of the replicating portfolio:B forward contract is freeB deposit will costINR2/1.05 INR1.905 Step 3 Law of One Price: option price valueportfolioC0 INR1.905

The Replication ApproachCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingThe Replication Approach Step 1 Replicate the payoff from the call—[5 if u] and[0 if d]:(a) forward contract(buy MTL 1/3 at 101)(b) deposit,V1 20(a) (b)S1 951/3 ( 95 101) 2 20S1 1101/3 (110 101) 3 25The Hedging ApproachThe Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial PricingTowardsBlackMertonScholes Step 2 Time-0 cost of the replicating portfolio:B forward contract is freeB deposit will costINR2/1.05 INR1.905 Step 3 Law of One Price: option price valueportfolioC0 INR1.905

The Hedging ApproachCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeReplication:Hedging: Step 1 Hedge the call(a) forward hdege(sell MTL 1/3 at 101)The Binomial Logic:One-period pricingThe Replication ApproachThe Hedging ApproachThe Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial PricingTowardsBlackMertonScholescall forward position riskfree depositcall – forward position riskfree deposit(b) call(a) (b)S1 951/3 (101 95) 202S1 1101/3 (101 110) 352 Step 2 time-0 value of the riskfree portfoliovalue INR 2/1.05 INR 1.905 Step 3 Law of one price: option price valueportfolioC0 [time-0 value of hedge] INR 1.905 C0 INR 1.905. otherwise there are arbitrage possibilities.

The Hedging ApproachCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeReplication:Hedging: Step 1 Hedge the call(a) forward hdege(sell MTL 1/3 at 101)The Binomial Logic:One-period pricingThe Replication ApproachThe Hedging ApproachThe Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial PricingTowardsBlackMertonScholescall forward position riskfree depositcall – forward position riskfree deposit(b) call(a) (b)S1 951/3 (101 95) 202S1 1101/3 (101 110) 352 Step 2 time-0 value of the riskfree portfoliovalue INR 2/1.05 INR 1.905 Step 3 Law of one price: option price valueportfolioC0 [time-0 value of hedge] INR 1.905 C0 INR 1.905. otherwise there are arbitrage possibilities.

The Hedging ApproachCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeReplication:Hedging: Step 1 Hedge the call(a) forward hdege(sell MTL 1/3 at 101)The Binomial Logic:One-period pricingThe Replication ApproachThe Hedging ApproachThe Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial PricingTowardsBlackMertonScholescall forward position riskfree depositcall – forward position riskfree deposit(b) call(a) (b)S1 951/3 (101 95) 202S1 1101/3 (101 110) 352 Step 2 time-0 value of the riskfree portfoliovalue INR 2/1.05 INR 1.905 Step 3 Law of one price: option price valueportfolioC0 [time-0 value of hedge] INR 1.905 C0 INR 1.905. otherwise there are arbitrage possibilities.

The Risk-adjusted ProbabilitiesCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingThe Replication ApproachThe Hedging Approach Overview: Implicitly, the replication/hedging story .B extracts a risk-adjusted probability “up” from the forwardmarket,The Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsB uses this probability to compute the call’s risk-adjustedexpected payoff, CEQ0 (C̃1 ); andStepwise MultiperiodBinomial PricingB discounts this risk-adjusted expectation at the riskfree rate.TowardsBlackMertonScholes

The Risk-adjusted ProbabilitiesCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingThe Replication ApproachThe Hedging Approach Overview: Implicitly, the replication/hedging story .B extracts a risk-adjusted probability “up” from the forwardmarket,The Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsB uses this probability to compute the call’s risk-adjustedexpected payoff, CEQ0 (C̃1 ); andStepwise MultiperiodBinomial PricingB discounts this risk-adjusted expectation at the riskfree rate.TowardsBlackMertonScholes

The Risk-adjusted ProbabilitiesCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingThe Replication ApproachThe Hedging Approach Overview: Implicitly, the replication/hedging story .B extracts a risk-adjusted probability “up” from the forwardmarket,The Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsB uses this probability to compute the call’s risk-adjustedexpected payoff, CEQ0 (C̃1 ); andStepwise MultiperiodBinomial PricingB discounts this risk-adjusted expectation at the riskfree rate.TowardsBlackMertonScholes

The Risk-adjusted ProbabilitiesCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingThe Replication ApproachThe Hedging Approach Step 1 Extract risk-adjusted probability from F:B Ordinary expectation:E0 (S̃1 ) p 110 (1 p) 95B Risk-adjusted expectation: CEQ0 (S̃1 ) q 110 (1 q) 95B We do not know how/why the market selects q, but q isrevealed by F0,1 ( 101):101 95 q (110 95) q The Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial Pricing101 956 0.4110 9515 Step 2 CEQ of the call’s payoff:CEQ0 (C̃1 ) (0.4 5) (1 0.4) 0 2TowardsBlackMertonScholes Step 3 Discount at r:C0 CEQ0 (C̃1 )2 1.9051 r0,11.05

The Risk-adjusted ProbabilitiesCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingThe Replication ApproachThe Hedging Approach Step 1 Extract risk-adjusted probability from F:B Ordinary expectation:E0 (S̃1 ) p 110 (1 p) 95B Risk-adjusted expectation: CEQ0 (S̃1 ) q 110 (1 q) 95B We do not know how/why the market selects q, but q isrevealed by F0,1 ( 101):101 95 q (110 95) q The Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial Pricing101 956 0.4110 9515 Step 2 CEQ of the call’s payoff:CEQ0 (C̃1 ) (0.4 5) (1 0.4) 0 2TowardsBlackMertonScholes Step 3 Discount at r:C0 CEQ0 (C̃1 )2 1.9051 r0,11.05

The Risk-adjusted ProbabilitiesCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingThe Replication ApproachThe Hedging Approach Step 1 Extract risk-adjusted probability from F:B Ordinary expectation:E0 (S̃1 ) p 110 (1 p) 95B Risk-adjusted expectation: CEQ0 (S̃1 ) q 110 (1 q) 95B We do not know how/why the market selects q, but q isrevealed by F0,1 ( 101):101 95 q (110 95) q The Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial Pricing101 956 0.4110 9515 Step 2 CEQ of the call’s payoff:CEQ0 (C̃1 ) (0.4 5) (1 0.4) 0 2TowardsBlackMertonScholes Step 3 Discount at r:C0 CEQ0 (C̃1 )2 1.9051 r0,11.05

The Risk-adjusted ProbabilitiesCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingThe Replication ApproachThe Hedging Approach Step 1 Extract risk-adjusted probability from F:B Ordinary expectation:E0 (S̃1 ) p 110 (1 p) 95B Risk-adjusted expectation: CEQ0 (S̃1 ) q 110 (1 q) 95B We do not know how/why the market selects q, but q isrevealed by F0,1 ( 101):101 95 q (110 95) q The Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial Pricing101 956 0.4110 9515 Step 2 CEQ of the call’s payoff:CEQ0 (C̃1 ) (0.4 5) (1 0.4) 0 2TowardsBlackMertonScholes Step 3 Discount at r:C0 CEQ0 (C̃1 )2 1.9051 r0,11.05

The Risk-adjusted ProbabilitiesCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingThe Replication ApproachThe Hedging Approach Step 1 Extract risk-adjusted probability from F:B Ordinary expectation:E0 (S̃1 ) p 110 (1 p) 95B Risk-adjusted expectation: CEQ0 (S̃1 ) q 110 (1 q) 95B We do not know how/why the market selects q, but q isrevealed by F0,1 ( 101):101 95 q (110 95) q The Risk-adjusted ProbabilitiesMultiperiod Pricing:AssumptionsStepwise MultiperiodBinomial Pricing101 956 0.4110 9515 Step 2 CEQ of the call’s payoff:CEQ0 (C̃1 ) (0.4 5) (1 0.4) 0 2TowardsBlackMertonScholes Step 3 Discount at r:C0 CEQ0 (C̃1 )2 1.9051 r0,11.05

OutlineCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingMultiperiod Pricing:AssumptionsThe Binomial Logic: One-period pricingThe Replication ApproachThe Hedging ApproachThe Risk-adjusted ProbabilitiesMultiperiod Pricing: sumptionsDiscussionStepwise MultiperiodBinomial PricingTowardsBlackMertonScholesStepwise Multiperiod Binomial Option PricingBackward Pricing, Dynamic HedgingWhat can go wrong?American-style OptionsTowards Black-Merton-ScholesSTP-ing of European OptionsTowards the Black-Merton-Scholes EquationThe Delta of an Option

Multiperiod Pricing: NotationCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingMultiperiod Pricing:Assumptions Subscripts: n,j whereB n says how many jumps have been made since time 0B j says how many of these jumps were up General pricing equation:Ct,j where qt NotationAssumptionsDiscussionStepwise MultiperiodBinomial PricingCt 1,u qt Ct 1,d (1 qt ),1 rt,1periodFt,t 1 St 1,d,St 1,u St 1,d1 r TowardsBlackMertonScholes dt St 1 rt,t 1 St dt t,t 1St ut St dt1 rt,t 1 1 rt,t 1, dtut dt,St 1,dSt 1,u, ut .StSt(1)

Multiperiod Pricing: NotationCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingMultiperiod Pricing:Assumptions Subscripts: n,j whereB n says how many jumps have been made since time 0B j says how many of these jumps were up General pricing equation:Ct,j where qt NotationAssumptionsDiscussionStepwise MultiperiodBinomial PricingCt 1,u qt Ct 1,d (1 qt ),1 rt,1periodFt,t 1 St 1,d,St 1,u St 1,d1 r TowardsBlackMertonScholes dt St 1 rt,t 1 St dt t,t 1St ut St dt1 rt,t 1 1 rt,t 1, dtut dt,St 1,dSt 1,u, ut .StSt(1)

AssumptionsCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingMultiperiod Pricing:AssumptionsNotationAssumptionsDiscussion A1 (r and r ) : The risk-free one-period rates of returnon both currencies are constantB denoted by unsubscripted r and r B Also assumed in Black-Scholes. A2 (u and d) : The multiplicative change factors, uand d, are constant.Also assumed in Black-Scholes:B no jumps (sudden de/revaluations) in the exchange rate process, andB a constant variance of the period-by-period percentage changes in S.Stepwise MultiperiodBinomial PricingTowardsBlackMertonScholes Implication of A1-A2: qt is a constant. A2.01 (no free lunch in F):d 1 r u St 1,d Ft St 1,u 0 q 11 r Q: what would you do if S1 [95 or 110] while F 90? 115?

AssumptionsCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingMultiperiod Pricing:AssumptionsNotationAssumptionsDiscussion A1 (r and r ) : The risk-free one-period rates of returnon both currencies are constantB denoted by unsubscripted r and r B Also assumed in Black-Scholes. A2 (u and d) : The multiplicative change factors, uand d, are constant.Also assumed in Black-Scholes:B no jumps (sudden de/revaluations) in the exchange rate process, andB a constant variance of the period-by-period percentage changes in S.Stepwise MultiperiodBinomial PricingTowardsBlackMertonScholes Implication of A1-A2: qt is a constant. A2.01 (no free lunch in F):d 1 r u St 1,d Ft St 1,u 0 q 11 r Q: what would you do if S1 [95 or 110] while F 90? 115?

AssumptionsCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingMultiperiod Pricing:AssumptionsNotationAssumptionsDiscussion A1 (r and r ) : The risk-free one-period rates of returnon both currencies are constantB denoted by unsubscripted r and r B Also assumed in Black-Scholes. A2 (u and d) : The multiplicative change factors, uand d, are constant.Also assumed in Black-Scholes:B no jumps (sudden de/revaluations) in the exchange rate process, andB a constant variance of the period-by-period percentage changes in S.Stepwise MultiperiodBinomial PricingTowardsBlackMertonScholes Implication of A1-A2: qt is a constant. A2.01 (no free lunch in F):d 1 r u St 1,d Ft St 1,u 0 q 11 r Q: what would you do if S1 [95 or 110] while F 90? 115?

AssumptionsCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeThe Binomial Logic:One-period pricingMultiperiod Pricing:AssumptionsNotationAssumptionsDiscussion A1 (r and r ) : The risk-free one-period rates of returnon both currencies are constantB denoted by unsubscripted r and r B Also assumed in Black-Scholes. A2 (u and d) : The multiplicative change factors, uand d, are constant.Also assumed in Black-Scholes:B no jumps (sudden de/revaluations) in the exchange rate process, andB a constant variance of the period-by-period percentage changes in S.Stepwise MultiperiodBinomial PricingTowardsBlackMertonScholes Implication of A1-A2: qt is a constant. A2.01 (no free lunch in F):d 1 r u St 1,d Ft St 1,u 0 q 11 r Q: what would you do if S1 [95 or 110] while F 90? 115?

950How such a tree worksO95Currency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeS3,3 S0uuuS2,2 S 0uuThe Binomial Logic:One-period pricingMultiperiod epwise MultiperiodBinomial PricingTowardsBlackMertonScholesS1,1 S0uS0S3,2 S0uudS2,1 S0udS1,0 S0dS3,1 S0uddS2,0 S0ddS3.0 S0 ddd11

The Emerging BellshapeCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPractice4.4 How to model (near-)impredictable spot rates (1)4. Time-series propertieslet p 1/2The Binomial Logic:One-period pricingMultiperiod epwise MultiperiodBinomial 83/83/81/81/16 n 4, j 4C 4!/4!0! 14/16 n 4, j 3C 4!/3!1! 24/6 46/16 n 4, j 2C 4!/2!2! 24/6 64/16 n 4, j 1C 4!/1!3! 24/6 41/16 n 4, j 0C 4!/4!0! 1The emerging bell-shape

! lnSN is normal ( S lognormal)WhatEmerging Why multiplicative? centsBellshape?vs percents no zero, negative S inverse of S Constant u and d: corresponds to constant % in BMSCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeadditive two oversimplifications:multiplicative Chosing betweenadditive10011090The Binomial Logic:One-period pricingMultiperiod 72.9– cents v percent:we prefer a constant distribution of percentage price changes over aconstant distribution of dollar price changes.– non-negative prices:with a multiplicative, the exchange rate can never quite reach zeroeven if it happens to go down all the time.DiscussionStepwise MultiperiodBinomial PricingTowardsBlackMertonScholes– invertible:we get a similar multiplicative process for the exchange rate as viewed abroad,S 1/S (with d 1/u, u 1/d). Corresponding Limiting Distributions:P t where { 10, 10}– additive: S̃n S0 nt 1 S̃n is normal if n is large (CLT)Q– multiplicative: S̃n S0 nt 1 (1 r̃t ) where r̃ { 10%, -10%}Pn ln S̃n ln S0 t 1 ρ̃t where ρ̃ ln(1 r̃) { 0.095, 0.095} ln S̃n is normal if n is large S̃n is lognormal.

! lnSN is normal ( S lognormal)WhatEmerging Why multiplicative? centsBellshape?vs percents no zero, negative S inverse of S Constant u and d: corresponds to constant % in BMSCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeadditive two oversimplifications:multiplicative Chosing betweenadditive10011090The Binomial Logic:One-period pricingMultiperiod 72.9– cents v percent:we prefer a constant distribution of percentage price changes over aconstant distribution of dollar price changes.– non-negative prices:with a multiplicative, the exchange rate can never quite reach zeroeven if it happens to go down all the time.DiscussionStepwise MultiperiodBinomial PricingTowardsBlackMertonScholes– invertible:we get a similar multiplicative process for the exchange rate as viewed abroad,S 1/S (with d 1/u, u 1/d). Corresponding Limiting Distributions:P t where { 10, 10}– additive: S̃n S0 nt 1 S̃n is normal if n is large (CLT)Q– multiplicative: S̃n S0 nt 1 (1 r̃t ) where r̃ { 10%, -10%}Pn ln S̃n ln S0 t 1 ρ̃t where ρ̃ ln(1 r̃) { 0.095, 0.095} ln S̃n is normal if n is large S̃n is lognormal.

! lnSN is normal ( S lognormal)WhatEmerging Why multiplicative? centsBellshape?vs percents no zero, negative S inverse of S Constant u and d: corresponds to constant % in BMSCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeadditive two oversimplifications:multiplicative Chosing betweenadditive10011090The Binomial Logic:One-period pricingMultiperiod 72.9– cents v percent:we prefer a constant distribution of percentage price changes over aconstant distribution of dollar price changes.– non-negative prices:with a multiplicative, the exchange rate can never quite reach zeroeven if it happens to go down all the time.DiscussionStepwise MultiperiodBinomial PricingTowardsBlackMertonScholes– invertible:we get a similar multiplicative process for the exchange rate as viewed abroad,S 1/S (with d 1/u, u 1/d). Corresponding Limiting Distributions:P t where { 10, 10}– additive: S̃n S0 nt 1 S̃n is normal if n is large (CLT)Q– multiplicative: S̃n S0 nt 1 (1 r̃t ) where r̃ { 10%, -10%}Pn ln S̃n ln S0 t 1 ρ̃t where ρ̃ ln(1 r̃) { 0.095, 0.095} ln S̃n is normal if n is large S̃n is lognormal.

! lnSN is normal ( S lognormal)WhatEmerging Why multiplicative? centsBellshape?vs percents no zero, negative S inverse of S Constant u and d: corresponds to constant % in BMSCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeadditive two oversimplifications:multiplicative Chosing betweenadditive10011090The Binomial Logic:One-period pricingMultiperiod 72.9– cents v percent:we prefer a constant distribution of percentage price changes over aconstant distribution of dollar price changes.– non-negative prices:with a multiplicative, the exchange rate can never quite reach zeroeven if it happens to go down all the time.DiscussionStepwise MultiperiodBinomial PricingTowardsBlackMertonScholes– invertible:we get a similar multiplicative process for the exchange rate as viewed abroad,S 1/S (with d 1/u, u 1/d). Corresponding Limiting Distributions:P t where { 10, 10}– additive: S̃n S0 nt 1 S̃n is normal if n is large (CLT)Q– multiplicative: S̃n S0 nt 1 (1 r̃t ) where r̃ { 10%, -10%}Pn ln S̃n ln S0 t 1 ρ̃t where ρ̃ ln(1 r̃) { 0.095, 0.095} ln S̃n is normal if n is large S̃n is lognormal.

! lnSN is normal ( S lognormal)WhatEmerging Why multiplicative? centsBellshape?vs percents no zero, negative S inverse of S Constant u and d: corresponds to constant % in BMSCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeadditive two oversimplifications:multiplicative Chosing betweenadditive10011090The Binomial Logic:One-period pricingMultiperiod 72.9– cents v percent:we prefer a constant distribution of percentage price changes over aconstant distribution of dollar price changes.– non-negative prices:with a multiplicative, the exchange rate can never quite reach zeroeven if it happens to go down all the time.DiscussionStepwise MultiperiodBinomial PricingTowardsBlackMertonScholes– invertible:we get a similar multiplicative process for the exchange rate as viewed abroad,S 1/S (with d 1/u, u 1/d). Corresponding Limiting Distributions:P t where { 10, 10}– additive: S̃n S0 nt 1 S̃n is normal if n is large (CLT)Q– multiplicative: S̃n S0 nt 1 (1 r̃t ) where r̃ { 10%, -10%}Pn ln S̃n ln S0 t 1 ρ̃t where ρ̃ ln(1 r̃) { 0.095, 0.095} ln S̃n is normal if n is large S̃n is lognormal.

! lnSN is normal ( S lognormal)WhatEmerging Why multiplicative? centsBellshape?vs percents no zero, negative S inverse of S Constant u and d: corresponds to constant % in BMSCurrency Options(2): Hedging andValuationP. Sercu,InternationalFinance: Theory intoPracticeadditive two oversimplifications:multiplicative Chosing betweenadditive10011090The Binomial Logic:One

Currency Options (2): Hedging and Valuation P. Sercu, International Finance: Theory into Practice Overview Overview The Binomial Logic: One-period pricing The Replication Approach The Hedging Approach The Risk-adjusted Probabilities Multiperiod Pricing: Assumptions Notation Assumptions Discussion Stepwise Multiperiod Binomial Option Pricing

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